Euler spiral

A double-end Euler spiral. The curve continues to converge to the points marked, as t tends to positive or negative infinity.

An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals.

Euler spirals have applications to diffraction computations. They are also widely used as transition curves in railroad engineering/highway engineering for connecting and transiting the geometry between a tangent and a circular curve. A similar application is also found in photonic integrated circuits. The principle of linear variation of the curvature of the transition curve between a tangent and a circular curve defines the geometry of the Euler spiral:

Its curvature begins with zero at the straight section (the tangent) and increases linearly with its curve length.

Where the Euler spiral meets the circular curve, its curvature becomes equal to that of the latter.

Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as its tip (click on thumbnail to observe).

To travel along a circular path, an object needs to be subject to a centripetal acceleration (eg: the moon circles around the earth because of gravity; a car turns its front wheels inward to generate a centripetal force). If a vehicle traveling on a straight path were to suddenly transition to a tangential circular path, it would require centripetal acceleration suddenly switching at the tangent point from zero to the required value; this would be difficult to achieve (think of a driver instantly moving the steering wheel from straight line to turning position, and the car actually doing it), putting mechanical stress on vehicle's parts, and causing much discomfort (causing jerk).

On early railroads this instant application of lateral force was not an issue since low speeds and wide-radius curves were employed (lateral forces on the passengers and the lateral sway was small and tolerable). As speeds of rail vehicles increased over the years, it became obvious that an easement is necessary so that the centripetal acceleration increases linearly with the traveled distance. Given the expression of centripetal acceleration V² / R, the obvious solution is to provide an easement curve whose curvature, 1 / R, increases linearly with the traveled distance. This geometry is an Euler spiral.

Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of degree 3), which is an approximation of the Euler spiral for small angular changes in the same way that a parabola is an approximation to a circular curve.

Marie Alfred Cornu (and later some civil engineers) also solved the calculus of Euler spiral independently. Euler spirals are now widely used in rail and highway engineering for providing a transition or an easement between a tangent and a horizontal circular curve.

Bends with continuously varying radius of curvature following the Euler spiral are also used to reduce losses in photonic integrated circuits, either in singlemode waveguides,[2][3] to smoothen the abrupt change of curvature and coupling to radiation modes, or in multimode waveguides,[4] in order to suppress coupling to higher order modes and ensure effective singlemode operation. A pioneering and very elegant application of the Euler spiral to waveguides had been made as early as 1957,[5] with a hollow metal waveguide for microwaves. There the idea was to exploit the fact that a straight metal waveguide can be physically bent to naturally take a gradual bend shape resembling an Euler spiral.

Angle of curve from beginning of spiral (infinite R{\displaystyle R}) to a particular point on the spiral.

This can also be measured as the angle between the initial tangent and the tangent at the concerned point.

θs{\displaystyle \theta _{s}}

Angle of full spiral curve

L,s{\displaystyle L,s}

Length measured along the spiral curve from its initial position

Ls,so{\displaystyle L_{s},s_{o}}

Length of spiral curve

Derivation

The graph on the right illustrates an Euler spiral used as an easement (transition) curve between two given curves, in this case a straight line (the negative x axis) and a circle. The spiral starts at the origin in the positive x direction and gradually turns anticlockwise to osculate the circle.

The spiral is a small segment of the above double-end Euler spiral in the first quadrant.

Generally the normalization reduces L' to a small value (<1) and results in good converging characteristics of the Fresnel integral manageable with only a few terms (at a price of increased numerical instability of the calculation, esp. for bigger θ{\displaystyle \theta } values.).

The two angles θs{\displaystyle \theta _{s}} are the same. This thus confirms that the original and normalized Euler spirals are geometrically similar. The locus of the normalized curve can be determined from Fresnel Integral, while the locus of the original Euler spiral can be obtained by scaling back / up or denormalizing.

The following SageMath code produces the second graph above. The first four lines express the Euler spiral component. Fresnel functions could not be found. Instead, the integrals of two expanded Taylor series are adopted. The remaining code expresses respectively the tangent and the circle, including the computation for the center coordinates.